Topological properties of curved spacetime extended Su-Schrieffer-Heeger model

This paper investigates the topological properties of an extended Su-Schrieffer-Heeger model in curved spacetime, demonstrating that while it retains the topological phases of its flat-space counterpart, it exhibits unique spatially asymmetric edge modes and critical slowdown phenomena near transition points that mimic gravitational horizons.

The Gist

Imagine you are walking through a perfectly flat, endless hallway. You walk at a constant speed, and if you drop a ball, it rolls straight ahead forever. This is how most physics models work: they assume space is flat and uniform.

Now, imagine that same hallway, but the floor gets sticky and rougher the closer you get to the left wall. As you walk toward that wall, you slow down. Eventually, you get so stuck that you seem to freeze in place, never quite reaching the wall, even though you keep trying. To an observer watching from far away, it looks like you've hit a "point of no return."

This is the basic idea behind black holes in our universe. Near a black hole's event horizon, gravity is so strong that nothing, not even light, can escape. In this paper, the authors built a tiny, artificial version of this "sticky hallway" using a model from condensed matter physics called the SSH model (named after Su, Schrieffer, and Heeger).

Here is the story of their discovery, broken down into simple concepts:

1. The "Magic" Hallway (The SSH Model)

The SSH model is like a chain of beads connected by springs. In the standard version, the springs are all the same strength. But this model is special because it can exist in two different "states":

• The Trivial State: The beads are just sitting there; nothing interesting happens at the ends.
• The Topological State: The beads at the very ends of the chain become "special." They hold onto energy in a way that makes them immune to defects or bumps in the chain. These are called edge states.

Think of the "Topological State" like a VIP lounge at the end of the hallway. Only the people at the very ends get to stay there, and they are protected from the chaos in the middle.

2. Introducing the "Gravity" (Curved Spacetime)

The authors asked: What happens if we make the springs in our bead chain get weaker and weaker as we get closer to the end, mimicking the pull of gravity near a black hole?

They introduced a "warping" factor (represented by the Greek letter sigma, $\sigma$).

• When $\sigma = 0$: The hallway is flat. The physics is normal.
• When $\sigma > 0$: The hallway curves. The "springs" (which represent how easily electrons hop between atoms) change strength depending on where you are.

3. The Big Discovery: The "Horizon" Appears

When they turned on this "gravity" (curved spacetime), they found something amazing:

• The Freeze: If they sent a wave of energy (a "wave packet") down the hallway toward the end, and the hallway was set to a specific "critical" setting, the wave would slow down and down and down. It would get closer and closer to the end but never actually reach it. It would take an infinite amount of time to get there.
• The Connection: This "eternal slowdown" is exactly what happens to light near a black hole's event horizon. The authors proved that you only see this "black hole" behavior when the system is at the exact moment it is switching between its two states (the Trivial and Topological phases).

It's as if the "VIP lounge" (the topological phase) only becomes a "black hole" exactly when the door is swinging open or closed.

4. The Asymmetry (The One-Way Mirror)

In the normal, flat version of this model, the "VIP lounges" at the left and right ends of the chain are identical twins. They behave exactly the same.

But in this "curved" version, the twins are no longer identical.

• The lounge on the right stays mostly the same.
• The lounge on the left (the side where the "gravity" is pulling) gets squashed and distorted. The energy gets stuck there much more tightly.
• If you try to move the wave from the left side, it bounces back and forth like a pinball before getting stuck, whereas the right side behaves more normally.

5. Why This Matters

Usually, scientists study black holes using giant telescopes or complex math about the universe. They also study topological materials (like new types of super-conductors) in the lab. These two fields rarely talk to each other.

This paper is a bridge. It shows that you can create a miniature black hole right inside a computer simulation of a simple chain of atoms.

• The "Event Horizon" isn't a place where you get crushed; it's a place where time seems to stop for a moving wave.
• The "Topological Protection" (the VIP status) survives even when the space is warped, proving that these quantum properties are incredibly robust.

The Takeaway

The authors discovered that topology (the shape of the quantum state) and gravity (curved spacetime) are deeply linked in these systems. By tweaking the "springs" in a simple model, they created a scenario where a particle slows down forever, mimicking a black hole, but only when the system is in a very specific, "critical" state.

It's like discovering that if you arrange your furniture in a room just right, the air in the corner will stop moving entirely, creating a tiny, invisible storm that traps everything that gets too close. This helps us understand both how black holes work and how to build better, more robust quantum computers.

Technical Summary

1. Problem Statement

The paper addresses the intersection of two distinct fields of physics: condensed matter topological systems and curved spacetime (CST) physics/black hole analogs.

• Context: While the Su-Schrieffer-Heeger (SSH) model is a well-understood 1D topological insulator in flat space, and various systems (acoustic, optical) have been used to simulate black hole horizons, the impact of curved spacetime geometry on the topological phases of such lattice models remains unexplored.
• Core Question: Can a topological system (specifically an extended SSH model) mimic event horizon physics? If so, how does the introduction of position-dependent hopping (simulating spacetime warping) affect topological invariants, edge states, and phase transitions?

2. Methodology

The authors propose and analyze a Curved Spacetime (CST) version of the Extended SSH model.

• Model Construction:

  • They define a tight-binding Hamiltonian with position-dependent hopping parameters ($t_n \propto (n/N)^\sigma$), where $\sigma$ represents the warping of spacetime.
  • The model includes nearest-neighbor ($t_1, t_2$) and next-nearest-neighbor ($t_3, t_4$) hoppings, allowing for higher-order topological phases (winding numbers $>1$).
  • In the continuum limit, this maps to a massless Dirac field in a Rindler metric ($ds^2 = -v^2(x)d\tau^2 + dx^2$), where $v(x) \to 0$ at the origin creates an effective event horizon.

• Analytical & Numerical Techniques:

  • Wave Packet Dynamics: Simulation of Gaussian wave packets to observe propagation near the origin (horizon).
  • Semiclassical Analysis: Derivation of semiclassical equations of motion ($\dot{x}, \dot{p}$) to predict turning points and critical slowdowns analytically.
  • Exact Diagonalization: Calculation of energy spectra and zero-energy eigenstates for finite systems.
  • Topological Markers: Since translational symmetry is broken by position-dependent hopping, momentum-space ($k$-space) winding numbers cannot be used. The authors employ real-space markers:
    • Local Topological Marker (LTM): A static measure of the winding number in the bulk.
    • Mean Chiral Displacement (MCD): A dynamical measure to detect winding numbers via time evolution.
  • Quench Dynamics: Studying the time evolution of survival probability when quenching from a topological phase to a trivial phase.
  • Inverse Participation Ratio (IPR): Used to quantify the localization of states near zero energy.

3. Key Contributions and Results

A. Emergence of Event Horizon Physics

• Critical Slowdown: The authors demonstrate that zero-energy wave packets propagating toward the origin (where hopping $t(x) \to 0$) undergo a critical slowdown.
• Horizon Condition: This slowdown becomes "eternal" (the packet never reaches the origin) only when the system parameters satisfy the topological phase transition conditions of the corresponding flat-space extended SSH model (i.e., where the bulk gap closes).
• Turning Points: For parameters away from the topological transition, wave packets reach a "turning point" ($x_{min}$) and bounce back. The analytical expression for $x_{min}$ confirms that $x_{min} \to 0$ only when the bulk gap closes in the homogeneous limit.

B. Topological Robustness in Curved Spacetime

• Symmetry Preservation: Despite the breaking of translational symmetry, the model retains Chiral and Time-Reversal symmetries, placing it in the BDI symmetry class with an integer $\mathbb{Z}$ topological invariant (winding number).
• Phase Transitions: Real-space markers (LTM and MCD) confirm that the CST-SSH model undergoes topological phase transitions at the exact same parameter values as the standard flat-space SSH model. The winding numbers remain robust against the spacetime warping ($\sigma$).

C. Asymmetric Edge States

• Localization Asymmetry: In the topological non-trivial phase, zero-energy edge states exist but are spatially asymmetric.
  • The edge state at the "horizon" side (origin) becomes increasingly localized as $\sigma$ increases.
  • The edge state at the opposite boundary remains relatively unchanged.
• Gap Scaling: The energy gap ($\Delta E$) scales as $N^{-\sigma}$.
  • For $\sigma < 1$, the system remains gapped in the thermodynamic limit.
  • For $\sigma \geq 1$, the spectrum becomes gapless. However, the authors show that gaplessness alone does not imply horizon physics; the critical slowdown only occurs at the specific topological transition points.

D. Mobility Gap and Localization

• Even when the spectrum is gapless ($\sigma \geq 1$), the states surrounding the zero-energy modes are spatially localized (confirmed by IPR calculations).
• This creates an effective mobility gap, preventing transport and maintaining the insulating nature of the system, which explains why the topological character is preserved despite the lack of a spectral gap.

E. Quench Dynamics and Oscillations

• When quenching from a topological phase to a trivial phase, the survival probability of edge states behaves differently depending on the side:
  • Right Edge (away from horizon): Dynamics are symmetric and similar to the flat SSH model.
  • Left Edge (near horizon): The survival probability exhibits oscillatory behavior (bouncing back and forth) rather than decaying monotonically. This is attributed to the wave packet being trapped near the horizon, moving slightly and reversing direction.

F. Quantification of Slowdown

• The authors define a characteristic time scale $\tau_s$ for the critical slowdown ($x(\tau) \propto e^{-\tau/\tau_s}$).
• They find that $\tau_s$ is maximized at higher-order gap-closing points (where multiple gap-closing conditions are met simultaneously), offering a new degree of freedom to control the "black hole" simulation in experimental setups.

4. Significance

• Theoretical Unification: The work successfully bridges the gap between topological condensed matter physics and analog gravity, proving that topological phase transitions in lattice models are the precise conditions required to generate synthetic event horizons.
• New Phenomenology: It reveals that topological edge states in curved spacetime are inherently asymmetric and that "horizon physics" (critical slowdown) is a phenomenon strictly tied to topological transitions, not just generic gaplessness.
• Experimental Relevance: The findings suggest that cold-atom experiments or photonic lattices can be tuned to simulate black hole horizons with high precision by targeting specific topological transition points. The ability to control the slowdown rate ($\tau_s$) via higher-order hopping parameters provides a novel experimental knob for studying Hawking radiation analogs and quantum dynamics near horizons.
• Robustness: It demonstrates that topological protection is robust even in the presence of strong spatial inhomogeneity (curved spacetime), provided the underlying symmetry class is preserved.